longitudinal wave
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Longitudinal waves are
wave In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. ''Periodic waves'' oscillate repeatedly about an equilibrium ...
s which
oscillate Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulu ...
in the direction which is parallel to the direction in which the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal waves are also called ''compressional'' or compression waves, because they produce compression and rarefaction when travelling through a medium, and pressure waves, because they produce increases and decreases in
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
. A wave along the length of a stretched Slinky toy, where the distance between coils increases and decreases, is a good visualization. Real-world examples include sound waves (vibrations in pressure, a particle of displacement, and particle velocity propagated in an
elastic Elastic is a word often used to describe or identify certain types of elastomer, Elastic (notion), elastic used in garments or stretch fabric, stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rub ...
medium) and seismic P waves (created by earthquakes and explosions). The other main type of wave is the transverse wave, in which the displacements of the medium are at right angles to the direction of propagation. Transverse waves, for instance, describe ''some'' bulk sound waves in
solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
materials (but not in
fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
s); these are also called " shear waves" to differentiate them from the (longitudinal) pressure waves that these materials also support.


Nomenclature

"Longitudinal waves" and "transverse waves" have been abbreviated by some authors as "L-waves" and "T-waves", respectively, for their own convenience. While these two abbreviations have specific meanings in
seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes (or generally, quakes) and the generation and propagation of elastic ...
(L-wave for Love wave or long wave) and
electrocardiography Electrocardiography is the process of producing an electrocardiogram (ECG or EKG), a recording of the heart's electrical activity through repeated cardiac cycles. It is an electrogram of the heart which is a graph of voltage versus time of t ...
(see
T wave In electrocardiography, the T wave represents the repolarization of the ventricles. The interval from the beginning of the QRS complex to the apex of the T wave is referred to as the ''absolute refractory period''. The last half of the T wav ...
), some authors chose to use "ℓ-waves" (lowercase 'L') and "t-waves" instead, although they are not commonly found in physics writings except for some popular science books.


Sound waves

For longitudinal harmonic sound waves, the
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
and
wavelength In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...
can be described by the formula :\ y(x,t) = y_\mathsf\cdot\cos\!\Bigl(\ \omega\cdot\left( t - \tfrac \right)\ \Bigr)\ where: : \ y\ ~~ is the displacement of the point on the traveling sound wave; : \ x\ ~~ is the distance from the point to the wave's source; : \ t\ ~~ is the time elapsed; : \ y_\mathsf\ is the amplitude of the oscillations, : \ c\ ~~ is the speed of the wave; and : \ \omega ~~ is the
angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
of the wave. The quantity \ \frac\ is the time that the wave takes to travel the distance \ x ~. The ordinary frequency (\ f\ ) of the wave is given by : f = \frac ~. The wavelength can be calculated as the relation between a wave's speed and ordinary frequency. : \lambda =\frac ~. For sound waves, the amplitude of the wave is the difference between the pressure of the undisturbed air and the maximum pressure caused by the wave. Sound's propagation speed depends on the type, temperature, and composition of the medium through which it propagates.


Speed of longitudinal waves


Isotropic medium

For
isotropic In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
solids and liquids, the speed of a longitudinal wave can be described by : \ v_\ell = \sqrt\ where : \ E_\ell\ ~~ is the
elastic modulus An elastic modulus (also known as modulus of elasticity (MOE)) is a quantity that describes an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. Definition The elastic modu ...
, such that \ E_\ell = K_b + \frac\ ::: where \ G\ ~~ is the shear modulus and \ K_b\ is the
bulk modulus The bulk modulus (K or B or k) of a substance is a measure of the resistance of a substance to bulk compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume. Other mo ...
; : \ \rho ~~~ is the mass
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
of the medium.


Attenuation of longitudinal waves

The
attenuation In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a Transmission medium, medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and ...
of a wave in a medium describes the loss of energy a wave carries as it propagates throughout the medium. This is caused by the scattering of the wave at interfaces, the loss of energy due to the friction between molecules, or geometric divergence. The study of attenuation of elastic waves in materials has increased in recent years, particularly within the study of polycrystalline materials where researchers aim to "nondestructively evaluate the degree of damage of engineering components" and to "develop improved procedures for characterizing microstructures" according to a research team led by R. Bruce Thompson in a '' Wave Motion'' publication.


Attenuation in viscoelastic materials

In
viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both Viscosity, viscous and Elasticity (physics), elastic characteristics when undergoing deformation (engineering), deformation. Viscous mate ...
materials, the attenuation coefficients per length \ \alpha_\ell\ for longitudinal waves and \ \alpha_T\ for transverse waves must satisfy the following ratio: :\ \frac ~\geq~ \frac\ where \ c_T\ and \ c_\ell\ are the transverse and longitudinal wave speeds respectively.


Attenuation in polycrystalline materials

Polycrystalline materials are made up of various crystal
grains A grain is a small, hard, dry fruit ( caryopsis) – with or without an attached hull layer – harvested for human or animal consumption. A grain crop is a grain-producing plant. The two main types of commercial grain crops are cereals and le ...
which form the bulk material. Due to the difference in crystal structure and properties of these grains, when a wave propagating through a poly-crystal crosses a grain boundary, a
scattering In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
event occurs causing scattering based attenuation of the wave. Additionally it has been shown that the ratio rule for viscoelastic materials, :\frac ~\geq~ \frac applies equally successfully to polycrystalline materials. A current prediction for modeling attenuation of waves in polycrystalline materials with elongated grains is the second-order approximation (SOA) model which accounts the second order of inhomogeneity allowing for the consideration multiple scattering in the crystal system. This model predicts that the shape of the grains in a poly-crystal has little effect on attenuation.


Pressure waves

The equations for sound in a fluid given above also apply to acoustic waves in an elastic solid. Although solids also support transverse waves (known as S-waves in
seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes (or generally, quakes) and the generation and propagation of elastic ...
), longitudinal sound waves in the solid exist with a
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
and wave impedance dependent on the material's
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
and its rigidity, the latter of which is described (as with sound in a gas) by the material's
bulk modulus The bulk modulus (K or B or k) of a substance is a measure of the resistance of a substance to bulk compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume. Other mo ...
. In May 2022, NASA reported the sonification (converting astronomical data associated with pressure waves into
sound In physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the br ...
) of the black hole at the center of the Perseus galaxy cluster.


Electromagnetics

Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
lead to the prediction of
electromagnetic wave In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength, ...
s in a vacuum, which are strictly transverse waves; due to the fact that they would need particles to vibrate upon, the electric and magnetic fields of which the wave consists are perpendicular to the direction of the wave's propagation. David J. Griffiths, Introduction to Electrodynamics, However plasma waves are longitudinal since these are not electromagnetic waves but density waves of charged particles, but which can couple to the electromagnetic field. After Heaviside's attempts to generalize Maxwell's equations, Heaviside concluded that electromagnetic waves were not to be found as longitudinal waves in "''
free space A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
''" or homogeneous media. Maxwell's equations, as we now understand them, retain that conclusion: in free-space or other uniform isotropic dielectrics, electro-magnetic waves are strictly transverse. However electromagnetic waves can display a longitudinal component in the electric and/or magnetic fields when traversing birefringent materials, or inhomogeneous materials especially at interfaces (surface waves for instance) such as Zenneck waves. In the development of modern physics, Alexandru Proca (1897–1955) was known for developing relativistic quantum field equations bearing his name (Proca's equations) which apply to the massive vector spin-1 mesons. In recent decades some other theorists, such as Jean-Pierre Vigier and Bo Lehnert of the Swedish Royal Society, have used the Proca equation in an attempt to demonstrate photon mass as a longitudinal electromagnetic component of Maxwell's equations, suggesting that longitudinal electromagnetic waves could exist in a Dirac polarized vacuum. However photon rest mass is strongly doubted by almost all physicists and is incompatible with the
Standard Model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
of physics.


See also

* Transverse wave *
Sound In physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the br ...
* Acoustic wave * P-wave * Plasma waves


References


Further reading

* Varadan, V. K., and Vasundara V. Varadan, "''Elastic wave scattering and propagation''". ''Attenuation due to scattering of ultrasonic compressional waves in granular media'' – A.J. Devaney, H. Levine, and T. Plona. Ann Arbor, Mich., Ann Arbor Science, 1982. * Schaaf, John van der, Jaap C. Schouten, and Cor M. van den Bleek, "''Experimental Observation of Pressure Waves in Gas-Solids Fluidized Beds''". American Institute of Chemical Engineers. New York, N.Y., 1997. * * * Russell, Dan, "''Longitudinal and Transverse Wave Motion''". Acoustics Animations, Pennsylvania State University, Graduate Program in Acoustics. * Longitudinal Waves, with animations "''The Physics Classroom''" {{Strings (music) Wave mechanics Articles containing video clips